📚SAT (Digital) Unit 1 – SAT Math – Algebra

Key Concepts and Foundations

• Understand the fundamental properties of real numbers (commutative, associative, distributive)
• Recognize and apply the order of operations (PEMDAS: parentheses, exponents, multiplication and division, addition and subtraction)
  • Remember that multiplication and division are performed from left to right, as are addition and subtraction
• Simplify algebraic expressions by combining like terms (terms with the same variables and exponents)
• Evaluate expressions by substituting values for variables
• Understand the concept of equality and the properties of equality (reflexive, symmetric, transitive)
  • These properties allow for the manipulation of equations to solve for unknown variables
• Identify and work with rational and irrational numbers
  • Rational numbers can be expressed as fractions or terminating/repeating decimals (0.5, 0.333...)
  • Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions ($\sqrt{2}$, $\pi$)
• Recognize and apply the properties of exponents (product rule, quotient rule, power rule, negative exponents)
• Understand the concept of absolute value and its properties (distance from zero on a number line)

Equations and Inequalities

• Solve linear equations in one variable using the addition, subtraction, multiplication, and division properties of equality
  • Isolate the variable by performing the same operation on both sides of the equation
• Solve equations with variables on both sides by combining like terms and using the properties of equality
• Solve equations involving fractions by multiplying both sides by the least common denominator (LCD)
• Solve equations with absolute value by considering the two possible cases (positive and negative)
• Solve quadratic equations using factoring, the quadratic formula, or completing the square
  • Factoring: $ax^2 + bx + c = 0$ can be solved by factoring the expression and setting each factor equal to zero
  • Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Solve systems of linear equations using substitution, elimination, or graphing
  • Substitution involves solving one equation for a variable and substituting the result into the other equation
  • Elimination involves multiplying one or both equations by a constant to eliminate a variable when the equations are added or subtracted
• Solve and graph linear inequalities in one variable
  • Use the properties of inequalities to isolate the variable, and remember to reverse the inequality sign when multiplying or dividing by a negative number
• Solve and graph systems of linear inequalities

Functions and Graphs

• Understand the concept of a function as a rule that assigns a unique output value to each input value
  • Use function notation $f(x)$ to represent the output value of a function for a given input value $x$
• Identify the domain (set of input values) and range (set of output values) of a function
• Recognize and work with linear, quadratic, exponential, and absolute value functions
  • Linear functions have the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept
  • Quadratic functions have the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$
  • Exponential functions have the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants, $a \neq 0$, and $b > 0$
• Interpret and analyze graphs of functions, including identifying intercepts, maxima, minima, and intervals of increase/decrease
• Understand the concept of slope as the rate of change of a linear function
  • Calculate slope using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ or $m = \frac{\text{rise}}{\text{run}}$
• Graph linear functions using the slope-intercept form $y = mx + b$ or the point-slope form $y - y_1 = m(x - x_1)$
• Transform functions by applying shifts, reflections, and dilations
  • Vertical shifts: $f(x) + k$; horizontal shifts: $f(x - h)$; reflections: $-f(x)$ or $f(-x)$; dilations: $a \cdot f(x)$ or $f(bx)$

Problem-Solving Strategies

• Read the problem carefully and identify the given information, the unknown, and the conditions or constraints
• Translate word problems into algebraic equations or inequalities
  • Assign variables to unknown quantities and use the given information to set up equations
• Break down complex problems into smaller, manageable steps
• Look for patterns or relationships that can simplify the problem or lead to a solution
• Consider alternative approaches or strategies if the initial approach is not successful
• Estimate or approximate solutions to check the reasonableness of the final answer
• Use logical reasoning and the process of elimination to narrow down answer choices in multiple-choice questions
• Draw diagrams, tables, or graphs to visualize the problem and organize the given information

Common Question Types

• Solving equations and inequalities in one variable
  • Linear equations, quadratic equations, equations with fractions or absolute values
• Solving systems of equations or inequalities
  • Systems of linear equations, systems of linear inequalities
• Analyzing and graphing functions
  • Identifying key features of graphs (intercepts, maxima, minima, intervals of increase/decrease)
  • Transforming functions (shifts, reflections, dilations)
• Word problems involving linear or quadratic relationships
  • Distance-rate-time problems, age problems, geometry problems (area, perimeter, volume)
• Evaluating expressions and functions
  • Substituting values for variables, using function notation
• Simplifying and manipulating algebraic expressions
  • Combining like terms, factoring, applying properties of exponents
• Interpreting and analyzing data from tables, graphs, or charts
  • Determining trends, making predictions, comparing quantities

Practice Problems and Solutions

• Work through a variety of practice problems that cover the key concepts and question types
  • Solve problems independently and then compare your solutions to the provided answers and explanations
• Analyze your mistakes and identify areas for improvement
  • Understand why your approach was incorrect and learn from the correct solution
• Focus on problems that involve multiple steps or require the application of multiple concepts
  • These problems will help develop your problem-solving skills and prepare you for more challenging questions on the test
• Practice translating word problems into algebraic equations or inequalities
  • Develop a systematic approach for assigning variables and setting up equations based on the given information
• Work on problems with different contexts and real-world applications
  • Exposure to a variety of contexts will improve your ability to recognize and apply algebraic concepts in different situations
• Practice solving problems under timed conditions to simulate the test-taking experience
  • This will help you develop time management skills and improve your efficiency in solving problems

Tips and Tricks for Test Day

• Review key formulas and concepts before the test
  • Focus on the main ideas and relationships rather than trying to memorize every detail
• Read each question carefully and identify the given information, the unknown, and any constraints
• Manage your time effectively
  • Don't spend too much time on any one question; if you get stuck, move on and come back later if time allows
• Show your work and write out the steps of your solution
  • This will help you organize your thoughts and make it easier to check your work or correct mistakes
• Use estimation and logical reasoning to eliminate answer choices that are clearly incorrect
• Double-check your answers, especially for questions that involve calculations
  • Make sure you have answered the question being asked and that your solution makes sense in the context of the problem
• If you finish the test early, use the remaining time to review your answers and make any necessary corrections
• Stay calm and confident throughout the test
  • Take deep breaths and remind yourself of your preparation and abilities

Additional Resources and Study Materials

• Official SAT practice tests and questions from the College Board
  • These materials provide the most accurate representation of the types of questions you will encounter on the actual test
• SAT preparation books from reputable publishers (Barron's, Kaplan, Princeton Review)
  • These books offer comprehensive review materials, practice problems, and test-taking strategies
• Online resources and tutorials (Khan Academy, IXL, Mathway)
  • These websites provide interactive lessons, practice problems, and video explanations for various algebraic concepts
• Study groups or tutoring sessions with peers or educators
  • Collaborating with others can help you learn from different perspectives and reinforce your understanding of the material
• Flashcards or summary sheets for key formulas, concepts, and definitions
  • Creating your own study aids can help you organize and retain important information
• Practice problems from textbooks or online sources
  • Exposure to a wide variety of problems will help you develop your problem-solving skills and prepare for the range of questions on the test